The modified Hartree–Fock self-consistent field equation

A one-electron correlation operator is introduced into the Hartree–Fock self-consistent field equation. The correlation operator is derived from the second-order perturbation theory. Energies of atomic and molecular systems calculated from this modified Hartree–Fock equation are equal to that from second-order perturbation of Hartree–Fock equation. The modified equation can also be solved self-consistently by the LCAO approximation. We also presented the modified expressions for other operators.     

Comparative study of the Hartree–Fock and the Hartree–Fock–Slater method

The Hartree–Fock method (standard Roothaan closed-shell HF –LCAO theory) and the Hartree–Fock–Slater method (restricted HFS –LCAO –DV method developed by Baerends and Ros) have been compared with emphasis on the respective one-electron equations and on the matrix elements of the respective Fock operators. Using the same STO basis in the two cases, the matrix elements of the Fock operators and of their separate one-electron, Coulomb, and exchange contributions have been calculated for the same orbitals and density of the ground state of the diatomic molecule ZnO. The effects of methodical (exchange potential) and numerical (DV method, density fit) differences between the HF and HFS methods on the various matrix elements have been analyzed. As expected the methodical effect prevails and is responsible for the higher (less negative) values of the matrix elements of the HFS Fock operator compared to those of the HF Fock operator. Numerical effects are observable also and are caused by the difference in integration procedures (DV method), not by the density fit.     

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.     

Numerical method for the open-shell restricted hartree–fock density-matrix direct calculation

A new computation procedure for direct calculation of the density matrix in the LCAO version of the restricted Hartree–Fock–Roothaan open-shell theory is analyzed. It is proved that the procedure is quadratically convergent and stable to the round-off errors independently of the Fock operator spectrum. The dependence of the limit matrix of the initial matrix is examined.     

Sur la theorie des equations de Hartree–Fock dependent du temps. V. Etude d'un modele spherique a 2 electrons. Calcul du tenseur d'hyperpolarisabilite d'ordre 3

In order to apply the TDHF equations in the matrix form we established previously, we conceive a model for which the static Hartree–Fock equations are rigorously solved; so we study problems only linked to the TDHF method, independent of any further approximation (for example, of LCAO type). This model is made of a spherical box containing electrons subject to a particular potential. We solve, numerically, the TDHF equations at different orders, and we calculate the third order hyperpolarizability tensor. We then compare our results with the ones given by a variation-perturbation method.     

Determination of ab initio polarizabilities of polymers: Application to polyethylene and polysilane

An ab initio method for calculating the longitudinal linear polarizability of polymeric chains is described. This method is equivalent to an uncoupled Hartree–Fock scheme. It is applied to polyethylene and polysilane in minimal STO-3G and extended 4-31G basis sets. The study describes important techniques for solving the difficulties met in actual calculations: band reordering of the band structures, calculation of analytical derivatives of the energy bands ?_n(k) and LCAO coefficients c_np(k), and errors caused by the improper lattice sum truncations of the Hartree–Fock matrix.     

Bond length alternation in cyclic polyenes. II. Unrestricted hartree–fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian, together with an empirical quasi harmonic σ-core potential is investigated using the unrestricted Hartree–Fock wave function employing different spatial orbitals for different spins. It is shown that in contrast to the restricted Hartree–Fock method, which favors bond alternation in large cyclic polyenes, the unrestricted Hartree–Fock method stabilizes the symmetric structures with equidistant internuclear separation. An assessment of the amount of correlation error recovered by the unrestricted Hartree–Fock procedure is examined and the qualitatively different behavior of the cyclic polyene models when described by restricted and unrestricted Hartree–Fock wave functions is discussed from this viewpoint.     

Bond length alternation in cyclic polyenes. I. Restricted Hartree–Fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian and its relationship to the singlet stability problem for symmetry adapted Hartree–Fock solutions for these systems is investigated using the restricted Hartree–Fock method. The σ-energy contribution is approximated by a quasiharmonic empirical potential. It is shown that the restricted Hartree–Fock energies favor the cyclic polyene distortion and an estimate of the distortion and of the stabilization energy for infinite linear polyenes is obtained.     

Density matrix of crystalline systems. I. Long-range behavior and related computational problems

The one-electron density matrix (DM ) of crystalline systems is discussed, especially concerning its longrange behavior; reference is made throughout to systems treated at a Hartree–Fock–LCAO –SCF level of approximation. The analysis is performed on the assumption of generally smooth behavior of eigenvalues and eigenvectors in k (reciprocal) space, so that they can be expressed by means of a truncated Fourier expansion. This assumption allows us to obtain analytic approximations for the DM , on the basis of the information collected at a few, suitably selected sampling k points. It is therefore possible at the same time to discuss the influence of structural parameters (dimensionality of the system, existence and shape of the Fermi surface, structure of the chemical bonds) and to set up a computational scheme that is general and simple enough.     

Some calculations with magnetic field dependent orbitals

Calculations with magnetic field dependent orbitals, which include usual cartesian gaussians and London's field dependent orbitals as special cases, are presented. Magnetic susceptibility and magnetic shielding of H₂ and HF molecules are calculated within the finite perturbation Hartree—Fock LCAO method.     

Some properties of the bivariational Hartree–Fock scheme for complex symmetric many-particle operators

The bivariational Hartree–Fock scheme for a general many-body operator T is discussed with particular reference to the complex symmetric case: T^? = T*. It shown that, even in the case when the complex symmetric operator T is real and hence also self-adjoint, the complex symmetric Hartree–Fock scheme does not reduce to the conventional real form, unless one introduces the constraint that the N-dimensional space spanned by the Hartree–Fock functions ? should be stable under complex conjugation, so that ?* = ?α. If one omits this constraint, one gets a complex symmetric formulation of the Hartree–Fock scheme for a real N-electron Hamiltonian having the properties H = H* = H^?, in which the effective Hamiltonian H_eff (1) may have complex eigenvalues ε_k. By using the method of complex scaling, it is indicated that these complex eigenvalues—at least for certain systems—may be related to the existence of so-called physical resonance states, and a simple example is given. Full details will be given elsewhere.     

An ab initio study of the unimolecular decomposition mechanism of formamidine. 4-31G Characterization of potential energy hypersurface

Ab initio MO calculations have been carried out for the unimolecular decomposition of formamidine. The Hartree–Fock method in LCAO approximation with the 4-31G basis set was used. The 4-31G potential hypersurface has been further studied. The stationary points (R, TS, and P) were localized. A reaction analysis by correlation of bond-order indices and localized molecular orbitals demonstrated that the decomposition is an asynchronous process. The TS can be described as four-membered ring.     

Simple derivation of conditions for instability in the Hartree–Fock and projected Hartree–Fock schemes

The conditions for instability of solutions of Hartree–Fock and projected Hartree–Fock equations are derived in a form involving finite real symmetric matrices. These conditions are also expressed in terms of the Fock–Dirac density matrix, both at the spin–orbital and at the orbital level. The particular variations which give rise to the so-called singlet and triplet instabilities are described.     

Highly correlated particle densities and idempotent one-densities

The problem of determining idempotent one-densities which integrate to the exact or to a highly correlated particle density is considered. A method for obtaining the minimum energy idempotent one-density integrating to a given correlated particle density within a finite basis is described. The implications of this are twofold. First, Hartree–Fock accuracy can be exceeded in describing the electron density with an idempotent one-density; this is particularly relevant to the problem of constructing orbitals from experimental x-ray scattering data. Second, electron densities from analytic CI or MCSCF wave functions can be made available in a form as compact as the Hartree–Fock density by reporting the orbitals which define the correlated density via an idempotent one-density. A numerical example of the new method is given in which an accurate correlated density for He is “fitted” by an idempotent one-density represented in a finite (near Hartree–Fock) basis. Considering the deficiencies of the basis for this purpose, a technique is suggested for constructing basis sets optimized for prediction of one-electron properties rather than for energy.     

Exact-exchange Hartree–Fock calculations for periodic systems. I. Illustration of the method

A scheme is presented for performing linear-combination-of-atomic-orbitals (LCAO ) self-consistent-field (SCF ) ab initio Hartree–Fock calculations of the electronic structure of periodic systems. The main aspects which characterize the present method are (i) a thorough discussion of both translational and local symmetry properties and the derivation of general formulas for the transformation of all the relevant monoelectronic and bielectronic terms under symmetry operators. (ii) The use of general yet powerful criteria for the truncation of infinite sums; in particular, the Coulomb electron–electron interactions are subdivided into terms corresponding to intersecting or nonintersecting charge distributions; the latter are grouped into shell contributions and the interaction is evaluated by multipolar expansions; the exchange interaction may be evaluated with great precision by retaining a relatively small number of two-electron integrals according to a truncation criterion which fully preserves its nonlocal character. (iii) The use of a procedure for performing integrals over k , as needed in the evaluation of the Fermi energy and in the reconstruction of the Fock matrix, which is particularly simple because it employs partially intersecting small spheres as integration subdomains where linear extrapolation is admitted. A comparison is finally made of our fundamental equations in the critical SCF stage with those obtainable by a recent proposal which uses Fourier transforms to express Coulomb and exchange integrals.     

An ab initio study of the hydrated positron

Ab initio calculations are presented for the hydration energy of the positron. Tetrahedral molecular-dipole-oriented clusters e⁺(H₂O)₄ are considered. In performing these calculations, the Hartree—Fock MO LCAO SCF approximation with the 4-31G split-valence basis set is used. The method was modified to treat the positron problem. It is shown that e⁺ in liquid water, like an electron, can be strongly solvated, with the hydration energy 0.2–0.3 eV greater than that of e⁺.     

Electronic correlation contribution to the three-body potentials for water trimers

A number of trimers of water molecules have been computed with an extended basis set in the Hartree–Fock and in the direct CI approximations. It has been verified that the three-body interaction energy can be calculated within the Hartree–Fock method. Therefore, the correlation corrections to the Hartree–Fock level are essentially additive and do not contribute significantly to three-body effects.     

Restricted hartree–fock method instability

The explicit forms of the spin density matrix variations which are responsible for the external instability in the restricted Hartree–Fock (RHF ) method are found. The RHF open shell instability matrix which guarantees the distorted wave function to belong to the same space as the initial one is derived for arbitrary spin.     

Corrections to coupled Hartree–Fock theory: the rearrangement effect

In this communication we discuss a method of incorporating corrections to the coupled Hartree–Fock (CHF ) formalism by introducing the so-called “rearrangement effect.” In this we take account of the relaxation of the core orbitals when excitations from a starting Hartree–Fock wave function occur. The magnitude of this correction numerically is found to be quite significant for the polarizabilities of two-electron atomic systems, results for which are reported.